The nonsymmetric eigenvalue problem is more complicated than the symmetric eigenvalue problem. In this subsection, we state the simplest bounds and leave the more complicated ones to subsequent subsections.
Let A be an n-by-n nonsymmetric matrix, with eigenvalues . Let vi be a right eigenvector corresponding to : . Let and be the corresponding computed eigenvalues and eigenvectors, computed by expert driver routine xGEEVX (see subsection 2.3.4).
The approximate error bound4.10for the computed eigenvalues are
EPSMCH = SLAMCH( 'E' ) * Compute the eigenvalues and eigenvectors of A * WR contains the real parts of the eigenvalues * WI contains the real parts of the eigenvalues * VL contains the left eigenvectors * VR contains the right eigenvectors CALL SGEEVX( 'P', 'V', 'V', 'B', N, A, LDA, WR, WI, $ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, $ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO ) IF( INFO.GT.0 ) THEN PRINT *,'SGEEVX did not converge' ELSE IF ( N.GT.0 ) THEN DO 10 I = 1, N EERRBD(I) = EPSMCH*ABNRM/RCONDE(I) VERRBD(I) = EPSMCH*ABNRM/RCONDV(I) 10 CONTINUE ENDIF
For example4.11, if